Here we provide the background of the invention in a less formalized language, stressing ideas rather than mathematical rigour. The purpose is to provide the big picture and the mathematical formalization of all concepts introduced here is given in subsection 5.1.
A computational approach to many scientific problems is often based on mathematical models idealizing how input data is transformed into output data. In this context, a forward problem can be characterized as a problem where the goal is to determine (or predict) the outcome when the models are applied to the input data, whereas in an inverse problem the goal is to determine the causes that, using the models, yields the output data.
Most inverse problems, especially the ones that we consider, can be formulated as the problem of solving an (operator) equation. More formally, assume that the object of interest in the inverse problem can be represented, or modeled, by an elements in a suitable set . Assume further that one cannot directly determine fε by measurements; instead one has a model of an experiment involving f describing how input data is transformed into output data. This model, derived using information applicable to the experiment, can be represented by an operator T that maps elements in  into another set , called the data space. In reality an experiment only yields finitely many, say m, noisy data points, so  is in this case an m-dimensional vector space. Hence, formally we have T: and the inverse problem can be stated as the problem of solving the equationT(f)=g where gεis given.The corresponding forward problem is to calculate g when f is given. To account for the stochasticity in the measurement process, one also needs to view data as a sample of some random variable and not as a fixed point. In this context, the inverse problem would be the problem to estimate fε from the data, which now is a single sample of some random element with values in the data space . A method that claims to (approximately) solve the inverse problem will be referred to as a reconstruction method.
A vast majority of inverse problems that are of interest are ill-posed. Intuitively, an inverse problem is well-posed (i.e. not ill-posed) if it has a unique solution for all (admissible) data and the solution depends continuously on the data. The last criterion simply ensures that the problem is stable, i.e. small errors in the data are not amplified. If one has non-attainable data then there are no solutions, i.e. we have non-existence. Next, there can be solutions but they are not unique, i.e. we have non-uniqueness. Finally, even if there is a unique solution, the process of inversion is not continuous, so the solution does not depend continuously on the data and we have instability. If a problem is ill-posed, non-existence is usually not the main concern. In fact, existence of a solution is an important requirement that can many times be achieved by modifying the problem as shown in subsection 5.1.2. Non-uniqueness is considered to be much more serious. The inverse problem with exact data has a unique solution whenever T is injective. In general, there is not much more to say about this case, although for a particular operator T it can of course be a difficult problem to determine if it is injective. The inverse problems occurring in practical applications, however, in general have finite data (so the data space  is finite dimensional), while the space  is infinite dimensional, and the forward operator can impossibly be injective. In this case one either has to choose one solution by some criteria or introduce uniqueness by adding additional information. This process is formalized in subsection 5.1.2. Finally, we have the case when one has instability, i.e. when the solution does not depend continuously on the data. This is the main reason for failure of many reconstruction methods that are based on numerically evaluating a discretized version of the inverse of the forward operator. Such an approach works well for well-posed problems, but if the inverse is not continuous, then one will experience numerical instabilities even when the data are only perturbed by small errors. This can partly be dealt with by the use of regularization methods which in general terms replaces an ill-posed problem by a family of neighboring well-posed problems. One has to keep in mind though that no mathematical trick can make an inherently ill-posed problem well-posed. All that a regularization method can do is to recover partial information about the solution in a stable manner. Thus, the “art” of applying regularization methods will always be to find the right compromise between accuracy and stability.
An important class of inverse problems are multicomponent inverse problems where the space  naturally splits into a number of components. This splitting of  can be used in deriving efficient reconstruction/regularization methods and regularization methods that take advantage of such a splitting is referred to here as component-wise regularization. An important example of a multicomponent inverse problem is blind deconvolution i.e. one seeks to de-convolve when the convolution kernel is unknown, so both the kernel and the function that is convolved is to be recovered. Another example is the identification problem in emission tomography where both the attenuation map and the activity map are unknown and needs to be reconstructed from measured data [8]. Finally, the inverse problem in electron tomography is an example of a multicomponent inverse problem where a number of experimental parameters must be recovered besides the scattering potential [5]. See also [7] for further examples from electrical engineering, medical and biological imaging, chemistry, robotics, vision, and environmental sciences.
Component-wise regularization/reconstruction, which is a method for dealing with multicomponent inverse problems (see subsection 5.1.6), has been described in the literature, see e.g. the survey article [7] that describes a component-wise non-linear least squares method. Iterated regularization methods have also been descried in the literature, see e.g. [12, 9, 2] for an analysis of iterated Tikhonov regularization. The combination as described here of component-wise regularization/reconstruction and iterated regularization/reconstruction is however not previously described.